4 



C. V. L. Charlier. 



of the objects iu question. We take into consideration a certain character of 

 these individuals, and assume that this character may be measured as to its 

 magnitude or intensity, so that the measurements are expressed through numbers. 

 Generally the character may vary continuously, and its true value in each indi- 

 vidual can then only be measured approximately as the height of a man. In 

 some cases the magnitude determinations of a character are expressed exactly in 

 numbers, as the numbers of petals in a flower. In either case we generally find 

 that the character varies from one individual to another. In known manner the 

 characters continuously varying may be treated in the same manner as those ex- 

 pressible in integers and we assume that, expressed in a certain unit, the character 

 X may assume all, or at least some, of the integer values 



0, ±1, ±2, +3, +4, ... 



Counting the number — // — of individuals having a certain magnitude in respect 

 to the character in consideration, we obtain what is called a frequency-table or — 

 graphically — a frequency-curve. 



What is the form of this curve? 



The question seems at the first glance to be somewhat vague, if not un- 

 answerable. Nevertheless experience has shown, that this curve really has a 

 certain form, which may be mathematically defined, and, what is still more astoni- 

 shing, that the parameters necessary to mathematically define a certain frequency- 

 curve arc generally very few in number. Very often 3 parameters suffice for repre- 

 senting, with satisfactory approximation, a collection of thousands of individuals. It 

 is the duty of the mathematician to find the equation of this curve. As to the 

 search for the hypotheses necessary to declare the origin of the frequency-curve, 

 the mathematician and the observer of the nature must work together. 



These hypotheses may be formulated in different ways. The question is 

 to find a hypothesis that will suffice for declaring all the different forms in which 

 the frequency-curves can occur. In searching for such a hypothesis we are aided 

 by the methods used in solving an astronomical problem of similar character. I 

 mean the explanation of the errors of observation. 



According to Hagen and Bessel, who have given the best explanation of this 

 difficult j-jroblem, an error of observation may be considered as the sum of a great 

 many very small elementary errors. Let us suppose the c|uestion is to determine the 

 siderial time through meridian observations of stars. If the transit instrument were 

 installed exactly iu the meridian, if the right ascensions of the stars were exactly 

 known, if the meteorological conditions of the atmosphere were known in all details, 

 if the physiological state of the observer at all observations were unaltered and if 

 all other circumstances that may have influence on the result were the same at 

 all observations, it is clear that we should obtain full agreement between the ob- 

 served values of the clock-correction. The true conditions, however, are somewhat 

 different from this ideal state. The adjustment of the instrument is not fufly correct, 



